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Stationary wave solutions for new developed twowaves’ fifthorder Korteweg–de Vries equation
Advances in Difference Equations volume 2019, Article number: 263 (2019)
Abstract
In this work, we present a new twowaves’ version of the fifthorder Korteweg–de Vries model. This model describes the propagation of moving twowaves under the influence of dispersion, nonlinearity, and phase velocity factors. We seek possible stationary wave solutions to this new model by means of Kudryashovexpansion method and sine–cosine function method. Also, we provide a graphical analysis to show the effect of phase velocity on the motion of the obtained solutions.
Introduction
Stationary wave solutions for nonlinear equations play an important role in understanding many mathematical models arising in physics and applied sciences. These solutions were developed and categorized to fit many physical learned aspects (see [1]). For example, the authors of [2,3,4] used rogue solitonwaves to study the coupled variablecoefficient fourthorder nonlinear Schrödinger equations in an inhomogeneous optical fiber and coupled Sasa–Satsuma equations. Hu et al. in [5] explored the mixed lump–kink and rogue wave–kink solutions for a \((3+1)\)dimensional Btype Kadomtsev–Petviashvili equation in fluid mechanics. Further, many interesting solitontype solutions for physical applications that arise in plasma, surface waves of finite depth, and optical fiber were studied by researchers in, e.g., [6,7,8,9].
In this work, we present a new interesting twowave version of the generalized fifthorder KdV equation which was discussed in [10, 11]. The standard fifthorder KdV equation has the form
where \(w=w(x,t)\) and \(a_{1}\), \(a_{2}\), \(a_{3}\) are some arbitrary constants. The fifthorder KdV equation (1.1) is a hybrid mathematical model with wide applications to surface and internal waves in fluids [11], as well as to waves in other media [12,13,14]. Special cases of (1.1) are widely used in different branches of sciences such as fluid physics, plasma physics, and quantum theory. For instance, when \(a_{1}=\frac{3}{10} a_{3}^{2}\), \(a_{2}=2 a _{3}\), \(a_{3}=10\), equation (1.1), called Lax equation, was studied in [15]. Also, when \(a_{1}=\frac{2}{5} a_{3}^{2}\), \(a_{2}=a _{3}\), \(a_{3}=5\), this equation is called Sawada–Kotera equation and was solved in [16]. In addition, the authors of [17] obtained the solution to (1.1) for \(a_{1}= \frac{1}{5}a_{3}^{2}\), \(a_{2}=a_{3}\), \(a_{3}=10\), which is known as the Kaup–Kupershmidt equation. Later on, under the assumption \(a_{1}=\frac{2}{9}a_{3}^{2}\), \(a_{2}=2a_{3}\), \(a_{3}=3\), the Ito equation was investigated in [18]. For the case \(a_{1}=45\), \(a_{2}= \frac{75}{2}\), \(a_{3}=15\), it is called Kaup–Kupershmidt–Parker–Dye equation [19]. The solution to Caudrey–Dodd–Gibbon equation was found in [20] provided that \(a_{1}=180\), \(a_{2}=30\), \(a _{3}=30\). Finally, for \(a_{1}=45\), \(a_{2}=15\), \(a_{3}=15\), (1.1) is called Sawada–Kotera–Parker–Dye equation, which was explored in [21].
The purpose of listing the aforementioned classifications of (1.1) is to highlight the importance and merit of studying new versions of the model, and also to explore its physical features. Now, we proceed to present for the first time the twowaves’ version of (1.1) by applying the operators
respectively, on the expressions \(a_{1} w^{2} w_{x}+a_{2} w_{x} w_{xx}+a _{3} w w_{xxx}\) and \(w_{xxxxx}\) and extending the term \(w_{t}\) into the expression \(w_{tt}s^{2} w_{xx}\). Therefore, the twowave fifthorder KdV (TWfKdV) is
where α, β, and s are the nonlinearity, dispersion, and phase velocity, respectively, with \(\alpha  \leq 1\), \(\beta  \leq 1\), and \(s \geq 0\). If we set \(s=0\) in (1.3) and integrate once with respect to time t, the TWfKdV equation is reduced to the fifthorder KdV equation (1.2) for the description of a singlewave propagating in one direction only. To learn about constructing twomode equations, the reader is advised to read [22,23,24,25,26,27,28,29,30,31].
The twowave equation (1.3) describes the spread of moving twowaves under the influence of dispersion, nonlinearity, and phase velocity factors. We aim to seek possible solutions for (1.3) by implementing two techniques, the Kudryashovexpansion method and sine–cosine function method. Also, we study the effect of phase velocity on the motion of the obtained solutions. Both techniques require converting (1.3) by means of the new variable \(\zeta =xc t\) into the differential equation
where \(w=w(\zeta )\).
Kudryashov solutions of TMfKdV
The Kudryashovexpansion technique [32,33,34,35] proposes the solution of (1.4) as a polynomial of the variable Z, namely
where variable Z satisfies the differential equation
Solving (2.2) gives
where d is a nonzero free constant. The index n is to be determined by applying the orderbalance procedure of the linear term \(w^{(5)}\) against the nonlinear term \(w^{2} w'\), which gives that \(n=2\). Therefore, we can write (2.1) as
Differentiating both (2.2) and (2.4) implicitly leads to
and
Now, we insert (2.2) through (2.6) into (1.4) to get a finite polynomial in Z. By setting each coefficient of \(Z^{i}\) to zero, a nonlinear algebraic system with unknowns \(A_{0}\), \(A_{1}\), \(A _{2}\), μ, c is obtained. We cannot solve the resulting system unless we consider some restrictions on the coefficients \(a_{0}\), \(a _{1}\), \(a_{2}\) and the parameters α, β.
KudryashovCase I
The first solution for the TMfKdV (1.3) exists when the coefficients are assigned as
and the twomode parameters have the relation
Hence,
Therefore, the first obtained solution is
Figure 1 presents 3D plots of the twowaves depicted in (2.8) upon increasing the phase velocity s. Figure 2 is a 2D plot of (2.8) when coordinate x is fixed. It can be seen that these two waves can be regarded as left–right waves (having opposite directions).
KudryashovCase II
When we take the coefficients
and the twowave parameters satisfy \(\alpha =\beta \), then the second solution for (1.3) is reached. Accordingly,
Thus, the second obtained solution is
Figure 3 presents the 3D plot of the twowaves depicted in (2.10).
KudryashovCase III
It is worth mentioning that when the twowaves’ parameters satisfy \(\alpha =\beta =\pm 1\), the third solution for TWfKdV (1.3) (with no restrictions on the coefficients \(a_{1}\), \(a_{2}\), \(a_{3}\)) is obtained. So,
which gives that the third obtained solution is
Figure 4 presents 3D plots of the twowaves depicted in (2.12) upon increasing the interaction phase velocity s.
Sine–cosine solution of TMfKdV
The goal of this section is to find periodic solutions of TWfKdV by means of sine–cosine function method (see [36,37,38]). This scheme propose the solution of (1.4) in the form
or
To determine the values of A, p, μ and c, we substitute (3.1) or (3.2) in (1.4), and then collect the coefficients of same powers of \(\sin ^{i}\) or \(\cos ^{i}\) and set each to zero. In fact, we have an algebraic system with \(\alpha =\beta \), namely
Solving (3.3) requires \(p=2\), and the TWfKdV’s coefficients are
So, we deduce that the wave speed c is
Therefore, two periodictype solutions are
Figure 5 presents plots of the twowaves obtained in (3.6) upon increasing the phase velocity s.
Conclusion
A new twowave version of the generalized fifthorder KdV problem is established. This new model possesses two directional waves with interacting phase velocity. We obtained different solutions of this new model under particular choices of the coefficients \(a_{1}\), \(a_{2}\), \(a _{3}\), and the constraint condition \(\alpha =\beta =d\) with \(d<1\). Also, we studied the impact of increasing the phase velocity on the shape of spreading its twowaves. The following findings are recorded:

For \(a_{1}=\frac{6 \mu ^{2} a_{3}}{A_{1}}\), \(a_{2}=\frac{60 \mu ^{2}A _{1} a_{3}}{A_{1}}\), \(a_{3}= \mathtt{free}\), and \(\alpha =\beta \), the TWfKdV is a solitontype.

For \(a_{1}=\frac{(183 7\sqrt{849}) \mu ^{4}}{8 A_{0}^{2}}\), \(a _{2}=\frac{(443 7\sqrt{849}) \mu ^{2}}{8 A_{0}}\), \(a_{3}= \frac{13 \mu ^{2}}{A_{0}}\), and \(\alpha =\beta \), the TWfKdV is a kinktype.

For arbitrary \(a_{1}\), \(a_{2}\), \(a_{3}\) and \(\alpha =\beta =\pm 1\), the TWfKdV is a kinktype.

For \(a_{1}=\frac{6 a_{3} \mu ^{2}}{A}\), \(a_{2}=\frac{A a_{3} +60 \mu ^{2}}{A}\) and \(\alpha =\beta \), the TWfKdV is a singular periodictype.
We may say that these twowaves could be useful in many physical and engineering applications, for example, they can be used as barrier waves to strengthen the transmission of different signals’ data. Also, if a large amount of data is difficult to pass on to a single router, it can be distributed on two routers.
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The authors would like to thank the editor and the anonymous referees for their indepth reading and insightful comments on an earlier version of this paper.
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Correspondence to Mohammed Ali.
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MSC
 35C08
 74J35
Keywords
 Twowave fifthorder KdV equation
 Kudryashovscheme
 Sine–cosine function method